Algebraic numbers

Definition

Algebraic numbers are numbers that are roots of non-zero polynomial equations with integer coefficients. They include rational numbers, which are also algebraic, as well as certain irrational numbers, such as the square root of 2. Understanding algebraic numbers is crucial for exploring classical construction problems and impossibility proofs, as these concepts often involve determining whether specific constructions can be achieved using only a compass and straightedge.

5 Must Know Facts For Your Next Test

1. Every rational number is an algebraic number since it can be expressed as a root of a linear polynomial like $ax + b = 0$.
2. Some well-known algebraic numbers include $rac{1}{2}$, $-rac{3}{4}$, and $ ext{sqrt}(3)$.
3. Transcendental numbers, like π and e, are not algebraic and cannot be constructed using classical methods.
4. Many classical construction problems, like doubling the cube or trisecting an angle, involve proving that certain numbers are not algebraic and hence cannot be constructed.
5. The set of algebraic numbers is countable, while the set of real numbers is uncountable, indicating that most real numbers are transcendental.

Review Questions

• How do algebraic numbers relate to classical construction problems involving compass and straightedge?
  • Algebraic numbers play a critical role in classical construction problems as they help determine whether certain geometric figures can be created using just a compass and straightedge. For example, many problems require proving that a specific length is algebraic to show that it can be constructed. If the required number is shown to be transcendental or not algebraic, then the construction is impossible.
• Discuss the significance of constructible numbers in relation to algebraic numbers in solving classical geometry problems.
  • Constructible numbers are a subset of algebraic numbers that specifically can be created through a series of geometric constructions. The significance lies in the fact that many classical problems hinge on whether the solutions involve constructible or non-constructible numbers. For instance, while all constructible numbers are algebraic, not all algebraic numbers are constructible. This distinction underlies several impossibility proofs in classical geometry.
• Evaluate the implications of recognizing the differences between algebraic and transcendental numbers in the context of impossibility proofs.
  • Recognizing the differences between algebraic and transcendental numbers has profound implications for impossibility proofs in geometry. It allows mathematicians to classify which constructions can be completed with traditional tools. For example, knowing that certain lengths derived from constructing roots of polynomials are not algebraic leads directly to the conclusion that those lengths cannot be achieved with compass and straightedge. This understanding not only enhances problem-solving strategies but also deepens insight into the limitations imposed by these fundamental mathematical concepts.

Related terms

rational numbers: Numbers that can be expressed as the quotient of two integers, where the denominator is not zero.

transcendental numbers: Numbers that are not algebraic, meaning they cannot be the root of any non-zero polynomial equation with integer coefficients, like π and e.

constructible numbers: Numbers that can be constructed from the integers using a finite number of operations involving addition, subtraction, multiplication, division, and square roots.